Comprehensive Study Notes
The time-dependent Schrödinger equation (1.13) is formidable looking. Fortunately, many applications of quantum mechanics to chemistry do not use this equation. Instead, the simpler time-independent Schrödinger equation is used. We now derive the
time-independent from the time-dependent Schrödinger equation for the one-particle, one-dimensional case.
We begin by restricting ourselves to the special case where the potential energy $V$ is not a function of time but depends only on $x$. This will be true if the system experiences no time-dependent external forces. The time-dependent Schrödinger equation reads
\(
\begin{equation*}
-\frac{\hbar}{i} \frac{\partial \Psi(x, t)}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}+V(x) \Psi(x, t) \tag{1.16}
\end{equation*}
\)
We now restrict ourselves to looking for those solutions of (1.16) that can be written as the product of a function of time and a function of $x$ :
\(
\begin{equation*}
\Psi(x, t)=f(t) \psi(x) \tag{1.17}
\end{equation*}
\)
Capital psi is used for the time-dependent wave function and lowercase psi for the factor that depends only on the coordinate $x$. States corresponding to wave functions of the form (1.17) possess certain properties (to be discussed shortly) that make them of great interest. [Not all solutions of (1.16) have the form (1.17); see Prob. 3.51.] Taking partial derivatives of (1.17), we have
\(
\frac{\partial \Psi(x, t)}{\partial t}=\frac{d f(t)}{d t} \psi(x), \quad \frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}=f(t) \frac{d^{2} \psi(x)}{d x^{2}}
\)
Substitution into (1.16) gives
\(
\begin{gather*}
-\frac{\hbar}{i} \frac{d f(t)}{d t} \psi(x)=-\frac{\hbar^{2}}{2 m} f(t) \frac{d^{2} \psi(x)}{d x^{2}}+V(x) f(t) \psi(x) \\
-\frac{\hbar}{i} \frac{1}{f(t)} \frac{d f(t)}{d t}=-\frac{\hbar^{2}}{2 m} \frac{1}{\psi(x)} \frac{d^{2} \psi(x)}{d x^{2}}+V(x) \tag{1.18}
\end{gather*}
\)
where we divided by $f \psi$. In general, we expect the quantity to which each side of (1.18) is equal to be a certain function of $x$ and $t$. However, the right side of (1.18) does not depend on $t$, so the function to which each side of (1.18) is equal must be independent of $t$. The left side of (1.18) is independent of $x$, so this function must also be independent of $x$. Since the function is independent of both variables, $x$ and $t$, it must be a constant. We call this constant $E$.
Equating the left side of (1.18) to $E$, we get
\(
\frac{d f(t)}{f(t)}=-\frac{i E}{\hbar} d t
\)
Integrating both sides of this equation with respect to $t$, we have
\(
\ln f(t)=-i E t / \hbar+C
\)
where $C$ is an arbitrary constant of integration. Hence
\(
f(t)=e^{C} e^{-i E t / \hbar}=A e^{-i E t / \hbar}
\)
where the arbitrary constant $A$ has replaced $e^{C}$. Since $A$ can be included as a factor in the function $\psi(x)$ that multiplies $f(t)$ in (1.17), $A$ can be omitted from $f(t)$. Thus
\(
f(t)=e^{-i E t / \hbar}
\)
Equating the right side of (1.18) to $E$, we have
\(
\begin{equation*}
-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi(x)}{d x^{2}}+V(x) \psi(x)=E \psi(x) \tag{1.19}
\end{equation*}
\)
Equation (1.19) is the time-independent Schrödinger equation for a single particle of mass $m$ moving in one dimension.
What is the significance of the constant $E$ ? Since $E$ occurs as $[E-V(x)]$ in (1.19), $E$ has the same dimensions as $V$, so $E$ has the dimensions of energy. In fact, we postulate that $E$ is the energy of the system. (This is a special case of a more general postulate to be discussed in a later chapter.) Thus, for cases where the potential energy is a function of $x$ only, there exist wave functions of the form
\(
\begin{equation*}
\Psi(x, t)=e^{-i E t / \hbar} \psi(x) \tag{1.20}
\end{equation*}
\)
and these wave functions correspond to states of constant energy $E$. Much of our attention in the next few chapters will be devoted to finding the solutions of (1.19) for various systems.
The wave function in (1.20) is complex, but the quantity that is experimentally observable is the probability density $|\Psi(x, t)|^{2}$. The square of the absolute value of a complex quantity is given by the product of the quantity with its complex conjugate, the complex conjugate being formed by replacing $i$ with $-i$ wherever it occurs. (See Section 1.7.) Thus
\(
\begin{equation*}
|\Psi|^{2}=\Psi^{*} \Psi \tag{1.21}
\end{equation*}
\)
where the star denotes the complex conjugate. For the wave function (1.20),
\(
\begin{align*}
|\Psi(x, t)|^{2} & =\left[e^{-i E t / \hbar} \psi(x)\right]^{*} e^{-i E t / \hbar} \psi(x) \\
& =e^{i E t / \hbar} \psi^{*}(x) e^{-i E t / \hbar} \psi(x) \\
& =e^{0} \psi^{*}(x) \psi(x)=\psi^{*}(x) \psi(x) \\
|\Psi(x, t)|^{2} & =|\psi(x)|^{2} \tag{1.22}
\end{align*}
\)
In deriving (1.22), we assumed that $E$ is a real number, so $E=E^{*}$. This fact will be proved in Section 7.2.
Hence for states of the form (1.20), the probability density is given by $|\Psi(x)|^{2}$ and does not change with time. Such states are called stationary states. Since the physically significant quantity is $|\Psi(x, t)|^{2}$, and since for stationary states $|\Psi(x, t)|^{2}=|\psi(x)|^{2}$, the function $\psi(x)$ is often called the wave function, although the complete wave function of a stationary state is obtained by multiplying $\psi(x)$ by $e^{-i E t / \hbar}$. The term stationary state should not mislead the reader into thinking that a particle in a stationary state is at rest. What is stationary is the probability density $|\Psi|^{2}$, not the particle itself.
We will be concerned mostly with states of constant energy (stationary states) and hence will usually deal with the time-independent Schrödinger equation (1.19). For simplicity we will refer to this equation as "the Schrödinger equation." Note that the Schrödinger equation contains two unknowns: the allowed energies $E$ and the allowed wave functions $\psi$. To solve for two unknowns, we need to impose additional conditions (called boundary conditions) on $\psi$ besides requiring that it satisfy (1.19). The boundary conditions determine the allowed energies, since it turns out that only certain values of $E$ allow $\psi$ to satisfy the boundary conditions. This will become clearer when we discuss specific examples in later chapters.
time-independent from the time-dependent Schrödinger equation for the one-particle, one-dimensional case.
We begin by restricting ourselves to the special case where the potential energy $V$ is not a function of time but depends only on $x$. This will be true if the system experiences no time-dependent external forces. The time-dependent Schrödinger equation reads
\(
\begin{equation*}
-\frac{\hbar}{i} \frac{\partial \Psi(x, t)}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}+V(x) \Psi(x, t) \tag{1.16}
\end{equation*}
\)
We now restrict ourselves to looking for those solutions of (1.16) that can be written as the product of a function of time and a function of $x$ :
\(
\begin{equation*}
\Psi(x, t)=f(t) \psi(x) \tag{1.17}
\end{equation*}
\)
Capital psi is used for the time-dependent wave function and lowercase psi for the factor that depends only on the coordinate $x$. States corresponding to wave functions of the form (1.17) possess certain properties (to be discussed shortly) that make them of great interest. [Not all solutions of (1.16) have the form (1.17); see Prob. 3.51.] Taking partial derivatives of (1.17), we have
\(
\frac{\partial \Psi(x, t)}{\partial t}=\frac{d f(t)}{d t} \psi(x), \quad \frac{\partial^{2} \Psi(x, t)}{\partial x^{2}}=f(t) \frac{d^{2} \psi(x)}{d x^{2}}
\)
Substitution into (1.16) gives
\(
\begin{gather*}
-\frac{\hbar}{i} \frac{d f(t)}{d t} \psi(x)=-\frac{\hbar^{2}}{2 m} f(t) \frac{d^{2} \psi(x)}{d x^{2}}+V(x) f(t) \psi(x) \\
-\frac{\hbar}{i} \frac{1}{f(t)} \frac{d f(t)}{d t}=-\frac{\hbar^{2}}{2 m} \frac{1}{\psi(x)} \frac{d^{2} \psi(x)}{d x^{2}}+V(x) \tag{1.18}
\end{gather*}
\)
where we divided by $f \psi$. In general, we expect the quantity to which each side of (1.18) is equal to be a certain function of $x$ and $t$. However, the right side of (1.18) does not depend on $t$, so the function to which each side of (1.18) is equal must be independent of $t$. The left side of (1.18) is independent of $x$, so this function must also be independent of $x$. Since the function is independent of both variables, $x$ and $t$, it must be a constant. We call this constant $E$.
Equating the left side of (1.18) to $E$, we get
\(
\frac{d f(t)}{f(t)}=-\frac{i E}{\hbar} d t
\)
Integrating both sides of this equation with respect to $t$, we have
\(
\ln f(t)=-i E t / \hbar+C
\)
where $C$ is an arbitrary constant of integration. Hence
\(
f(t)=e^{C} e^{-i E t / \hbar}=A e^{-i E t / \hbar}
\)
where the arbitrary constant $A$ has replaced $e^{C}$. Since $A$ can be included as a factor in the function $\psi(x)$ that multiplies $f(t)$ in (1.17), $A$ can be omitted from $f(t)$. Thus
\(
f(t)=e^{-i E t / \hbar}
\)
Equating the right side of (1.18) to $E$, we have
\(
\begin{equation*}
-\frac{\hbar^{2}}{2 m} \frac{d^{2} \psi(x)}{d x^{2}}+V(x) \psi(x)=E \psi(x) \tag{1.19}
\end{equation*}
\)
Equation (1.19) is the time-independent Schrödinger equation for a single particle of mass $m$ moving in one dimension.
What is the significance of the constant $E$ ? Since $E$ occurs as $[E-V(x)]$ in (1.19), $E$ has the same dimensions as $V$, so $E$ has the dimensions of energy. In fact, we postulate that $E$ is the energy of the system. (This is a special case of a more general postulate to be discussed in a later chapter.) Thus, for cases where the potential energy is a function of $x$ only, there exist wave functions of the form
\(
\begin{equation*}
\Psi(x, t)=e^{-i E t / \hbar} \psi(x) \tag{1.20}
\end{equation*}
\)
and these wave functions correspond to states of constant energy $E$. Much of our attention in the next few chapters will be devoted to finding the solutions of (1.19) for various systems.
The wave function in (1.20) is complex, but the quantity that is experimentally observable is the probability density $|\Psi(x, t)|^{2}$. The square of the absolute value of a complex quantity is given by the product of the quantity with its complex conjugate, the complex conjugate being formed by replacing $i$ with $-i$ wherever it occurs. (See Section 1.7.) Thus
\(
\begin{equation*}
|\Psi|^{2}=\Psi^{*} \Psi \tag{1.21}
\end{equation*}
\)
where the star denotes the complex conjugate. For the wave function (1.20),
\(
\begin{align*}
|\Psi(x, t)|^{2} & =\left[e^{-i E t / \hbar} \psi(x)\right]^{*} e^{-i E t / \hbar} \psi(x) \\
& =e^{i E t / \hbar} \psi^{*}(x) e^{-i E t / \hbar} \psi(x) \\
& =e^{0} \psi^{*}(x) \psi(x)=\psi^{*}(x) \psi(x) \\
|\Psi(x, t)|^{2} & =|\psi(x)|^{2} \tag{1.22}
\end{align*}
\)
In deriving (1.22), we assumed that $E$ is a real number, so $E=E^{*}$. This fact will be proved in Section 7.2.
Hence for states of the form (1.20), the probability density is given by $|\Psi(x)|^{2}$ and does not change with time. Such states are called stationary states. Since the physically significant quantity is $|\Psi(x, t)|^{2}$, and since for stationary states $|\Psi(x, t)|^{2}=|\psi(x)|^{2}$, the function $\psi(x)$ is often called the wave function, although the complete wave function of a stationary state is obtained by multiplying $\psi(x)$ by $e^{-i E t / \hbar}$. The term stationary state should not mislead the reader into thinking that a particle in a stationary state is at rest. What is stationary is the probability density $|\Psi|^{2}$, not the particle itself.
We will be concerned mostly with states of constant energy (stationary states) and hence will usually deal with the time-independent Schrödinger equation (1.19). For simplicity we will refer to this equation as "the Schrödinger equation." Note that the Schrödinger equation contains two unknowns: the allowed energies $E$ and the allowed wave functions $\psi$. To solve for two unknowns, we need to impose additional conditions (called boundary conditions) on $\psi$ besides requiring that it satisfy (1.19). The boundary conditions determine the allowed energies, since it turns out that only certain values of $E$ allow $\psi$ to satisfy the boundary conditions. This will become clearer when we discuss specific examples in later chapters.